 
Summary: A New Category for Semantics
Andrej Bauer and Dana Scott
June 2001
Domain theory for denotational semantics is over thirty years old. There are
many variations on the idea and many interesting constructs that have been
proposed by many people for realizing a wide variety of types as domains.
Generally, the effort has been to create categories of domains that are carte
sian closed (that is, have products and function spaces interpreting typed
calculus) and permit solutions to domain equations (that is, interpret recursive
domain definitions and perhaps untyped calculus).
What has been missing is a simple connection between domains and the
usual settheoretical structures of mathematics as well as a comprehensive
logic to reason about domains and the functions to be defined upon them. In
December of 1996, Scott realized that the very old idea of partial equivalence
relations on types could be applied to produce a large and rich category con
taining many specific categories of domains and allowing a suitable general
logic. The category is called Equ, the category of equilogical spaces.
The simplest definition is the category of T0spaces and total equivalence
relations with continuous maps that are equivariant (meaning, preserving the
equivalence relations). An equivalent definition uses algebraic (or continuous)
